I’m sorry to ask for help again , but I can’t prove the following identity:

$\displaystyle \dfrac{\tan^3\theta}{1 + \tan^2\theta} + \dfrac{\cot^3\theta}{1 + \cot^2\theta} \equiv \dfrac{1 -2\sin^2\theta\cos^2\theta}{\sin\theta\cos\theta}$

I have tried but I’m afraid I simply can’t do it.

I keep getting:

$\displaystyle LHS \equiv \dfrac{\sin^4\theta + cos^4\theta}{\sin\theta\cos\theta}$

And I can’t see how to equate the numerators. I think I've probably made a mistake.

A hint would be very much appreciated.

Thank you.